xrge0iifiso - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0iifiso		Structured version Visualization version GIF version

Theorem xrge0iifiso 33881

Description: The defined bijection from the closed unit interval onto the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)

**Hypothesis**
Ref	Expression
xrge0iifhmeo.1	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))

**Assertion**
Ref	Expression
xrge0iifiso	⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))

Distinct variable group: 𝑥,𝐹

Proof of Theorem xrge0iifiso Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 13490	. . 3 ⊢ (0[,]1) ⊆ ℝ^*
2		xrltso 13203	. . 3 ⊢ < Or ℝ^*
3		soss 5628	. . 3 ⊢ ((0[,]1) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]1)))
4	1, 2, 3	mp2 9	. 2 ⊢ < Or (0[,]1)
5		iccssxr 13490	. . 3 ⊢ (0[,]+∞) ⊆ ℝ^*
6		cnvso 6319	. . . . 5 ⊢ ( < Or ℝ^* ↔ ^◡ < Or ℝ^*)
7	2, 6	mpbi 230	. . . 4 ⊢ ^◡ < Or ℝ^*
8		sopo 5627	. . . 4 ⊢ (^◡ < Or ℝ^* → ^◡ < Po ℝ^*)
9	7, 8	ax-mp 5	. . 3 ⊢ ^◡ < Po ℝ^*
10		poss 5609	. . 3 ⊢ ((0[,]+∞) ⊆ ℝ^* → (^◡ < Po ℝ^* → ^◡ < Po (0[,]+∞)))
11	5, 9, 10	mp2 9	. 2 ⊢ ^◡ < Po (0[,]+∞)
12		xrge0iifhmeo.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
13	12	xrge0iifcnv 33879	. . . 4 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑧 ∈ (0[,]+∞) ↦ if(𝑧 = +∞, 0, (exp‘-𝑧))))
14	13	simpli 483	. . 3 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
15		f1ofo 6869	. . 3 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) → 𝐹:(0[,]1)–onto→(0[,]+∞))
16	14, 15	ax-mp 5	. 2 ⊢ 𝐹:(0[,]1)–onto→(0[,]+∞)
17		0xr 11337	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
18		1xr 11349	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
19		0le1 11813	. . . . . . . 8 ⊢ 0 ≤ 1
20		snunioc 13540	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
21	17, 18, 19, 20	mp3an 1461	. . . . . . 7 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
22	21	eleq2i 2836	. . . . . 6 ⊢ (𝑤 ∈ ({0} ∪ (0(,]1)) ↔ 𝑤 ∈ (0[,]1))
23		elun 4176	. . . . . 6 ⊢ (𝑤 ∈ ({0} ∪ (0(,]1)) ↔ (𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)))
24	22, 23	bitr3i 277	. . . . 5 ⊢ (𝑤 ∈ (0[,]1) ↔ (𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)))
25		velsn 4664	. . . . . . 7 ⊢ (𝑤 ∈ {0} ↔ 𝑤 = 0)
26		elunitrn 13527	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ∈ ℝ)
27	26	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ∈ ℝ)
28		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 0 < 𝑧)
29		elicc01 13526	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
30	29	simp3bi 1147	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ≤ 1)
31	30	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ≤ 1)
32		1re 11290	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
33		elioc2 13470	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑧 ∈ (0(,]1) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ 1)))
34	17, 32, 33	mp2an 691	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ 1))
35	27, 28, 31, 34	syl3anbrc 1343	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ∈ (0(,]1))
36		pnfxr 11344	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
37		0le0 12394	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 0
38		ltpnf 13183	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℝ → 1 < +∞)
39	32, 38	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 1 < +∞
40		iocssioo 13499	. . . . . . . . . . . . . . 15 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
41	17, 36, 37, 39, 40	mp4an 692	. . . . . . . . . . . . . 14 ⊢ (0(,]1) ⊆ (0(,)+∞)
42		ioorp 13485	. . . . . . . . . . . . . 14 ⊢ (0(,)+∞) = ℝ⁺
43	41, 42	sseqtri 4045	. . . . . . . . . . . . 13 ⊢ (0(,]1) ⊆ ℝ⁺
44	43	sseli 4004	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0(,]1) → 𝑧 ∈ ℝ⁺)
45		relogcl 26635	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → (log‘𝑧) ∈ ℝ)
46	45	renegcld 11717	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ⁺ → -(log‘𝑧) ∈ ℝ)
47		ltpnf 13183	. . . . . . . . . . . . . 14 ⊢ (-(log‘𝑧) ∈ ℝ → -(log‘𝑧) < +∞)
48	46, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ⁺ → -(log‘𝑧) < +∞)
49		brcnvg 5904	. . . . . . . . . . . . . 14 ⊢ ((+∞ ∈ ℝ^* ∧ -(log‘𝑧) ∈ ℝ) → (+∞^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < +∞))
50	36, 46, 49	sylancr 586	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ⁺ → (+∞^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < +∞))
51	48, 50	mpbird 257	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → +∞^◡ < -(log‘𝑧))
52	44, 51	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) → +∞^◡ < -(log‘𝑧))
53	12	xrge0iifcv 33880	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) → (𝐹‘𝑧) = -(log‘𝑧))
54	52, 53	breqtrrd 5194	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0(,]1) → +∞^◡ < (𝐹‘𝑧))
55	35, 54	syl 17	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → +∞^◡ < (𝐹‘𝑧))
56	55	ex 412	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,]1) → (0 < 𝑧 → +∞^◡ < (𝐹‘𝑧)))
57		breq1 5169	. . . . . . . . 9 ⊢ (𝑤 = 0 → (𝑤 < 𝑧 ↔ 0 < 𝑧))
58		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑤 = 0 → (𝐹‘𝑤) = (𝐹‘0))
59		0elunit 13529	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]1)
60		iftrue 4554	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
61		pnfex 11343	. . . . . . . . . . . . 13 ⊢ +∞ ∈ V
62	60, 12, 61	fvmpt 7029	. . . . . . . . . . . 12 ⊢ (0 ∈ (0[,]1) → (𝐹‘0) = +∞)
63	59, 62	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹‘0) = +∞
64	58, 63	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝑤 = 0 → (𝐹‘𝑤) = +∞)
65	64	breq1d 5176	. . . . . . . . 9 ⊢ (𝑤 = 0 → ((𝐹‘𝑤)^◡ < (𝐹‘𝑧) ↔ +∞^◡ < (𝐹‘𝑧)))
66	57, 65	imbi12d 344	. . . . . . . 8 ⊢ (𝑤 = 0 → ((𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)) ↔ (0 < 𝑧 → +∞^◡ < (𝐹‘𝑧))))
67	56, 66	imbitrrid 246	. . . . . . 7 ⊢ (𝑤 = 0 → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
68	25, 67	sylbi 217	. . . . . 6 ⊢ (𝑤 ∈ {0} → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
69		simpll 766	. . . . . . . . 9 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 ∈ (0(,]1))
70	26	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ∈ ℝ)
71		0re 11292	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
72	71	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 ∈ ℝ)
73	43	sseli 4004	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0(,]1) → 𝑤 ∈ ℝ⁺)
74	73	rpred 13099	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0(,]1) → 𝑤 ∈ ℝ)
75	74	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 ∈ ℝ)
76		elioc2 13470	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑤 ∈ (0(,]1) ↔ (𝑤 ∈ ℝ ∧ 0 < 𝑤 ∧ 𝑤 ≤ 1)))
77	17, 32, 76	mp2an 691	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0(,]1) ↔ (𝑤 ∈ ℝ ∧ 0 < 𝑤 ∧ 𝑤 ≤ 1))
78	77	simp2bi 1146	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0(,]1) → 0 < 𝑤)
79	78	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 < 𝑤)
80		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 < 𝑧)
81	72, 75, 70, 79, 80	lttrd 11451	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 < 𝑧)
82	30	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ≤ 1)
83	70, 81, 82, 34	syl3anbrc 1343	. . . . . . . . 9 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ∈ (0(,]1))
84	69, 83	jca 511	. . . . . . . 8 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → (𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)))
85	73	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → 𝑤 ∈ ℝ⁺)
86	85	relogcld 26683	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (log‘𝑤) ∈ ℝ)
87	44	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → 𝑧 ∈ ℝ⁺)
88	87	relogcld 26683	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (log‘𝑧) ∈ ℝ)
89	86, 88	ltnegd 11868	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ((log‘𝑤) < (log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤)))
90		logltb 26660	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺) → (𝑤 < 𝑧 ↔ (log‘𝑤) < (log‘𝑧)))
91	73, 44, 90	syl2an 595	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 ↔ (log‘𝑤) < (log‘𝑧)))
92		negex 11534	. . . . . . . . . . . . 13 ⊢ -(log‘𝑤) ∈ V
93		negex 11534	. . . . . . . . . . . . 13 ⊢ -(log‘𝑧) ∈ V
94	92, 93	brcnv 5907	. . . . . . . . . . . 12 ⊢ (-(log‘𝑤)^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤))
95	94	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (-(log‘𝑤)^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤)))
96	89, 91, 95	3bitr4d 311	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 ↔ -(log‘𝑤)^◡ < -(log‘𝑧)))
97	96	biimpd 229	. . . . . . . . 9 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 → -(log‘𝑤)^◡ < -(log‘𝑧)))
98	12	xrge0iifcv 33880	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0(,]1) → (𝐹‘𝑤) = -(log‘𝑤))
99	98, 53	breqan12d 5182	. . . . . . . . 9 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ((𝐹‘𝑤)^◡ < (𝐹‘𝑧) ↔ -(log‘𝑤)^◡ < -(log‘𝑧)))
100	97, 99	sylibrd 259	. . . . . . . 8 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))
101	84, 80, 100	sylc 65	. . . . . . 7 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))
102	101	exp31 419	. . . . . 6 ⊢ (𝑤 ∈ (0(,]1) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
103	68, 102	jaoi 856	. . . . 5 ⊢ ((𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
104	24, 103	sylbi 217	. . . 4 ⊢ (𝑤 ∈ (0[,]1) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
105	104	imp 406	. . 3 ⊢ ((𝑤 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))
106	105	rgen2 3205	. 2 ⊢ ∀𝑤 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))
107		soisoi 7364	. 2 ⊢ ((( < Or (0[,]1) ∧ ^◡ < Po (0[,]+∞)) ∧ (𝐹:(0[,]1)–onto→(0[,]+∞) ∧ ∀𝑤 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))) → 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞)))
108	4, 11, 16, 106, 107	mp4an 692	1 ⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∪ cun 3974 ⊆ wss 3976 ifcif 4548 {csn 4648 class class class wbr 5166 ↦ cmpt 5249 Po wpo 5605 Or wor 5606 ^◡ccnv 5699 –onto→wfo 6571 –1-1-onto→wf1o 6572 ‘cfv 6573 Isom wiso 6574 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 +∞cpnf 11321 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 -cneg 11521 ℝ⁺crp 13057 (,)cioo 13407 (,]cioc 13408 [,]cicc 13410 expce 16109 logclog 26614

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616

This theorem is referenced by: xrge0iifhmeo 33882

W3C validator